Exact extended formulation of the linear assignment problem (LAP) polytope for solving the traveling salesman and quadratic assignment problems

Abstract

We present an O(n 6 ) linear programming model for the traveling salesman (TSP) and quadratic assignment (QAP) problems. The basic model is developed within the framework of the TSP. It does not involve the city-to-city variables-based, traditional TSP polytope referred to in the literature as "the TSP polytope." We do not model explicit Hamiltonian cycles of the cities. Instead, we use a time-dependent abstraction of TSP tours and develop a direct extended formulation of the linear assignment problem (LAP) polytope. The model is exact in the sense that it has integral extreme points which are in one-to-one correspondence with TSP tours. It can be solved optimally using any linear programming (LP) solver, hence offering a new (incidental) proof of the equality of the computational complexity classes "P " and "N P ." The extensions of the model to the time-dependent traveling salesman problem (TDTSP) as well as the quadratic assignment problem (QAP) are straightforward. The reasons for the non-applicability of existing negative extended formulations results for "the TSP polytope" to the model in this paper as well as our software implementation and the computational experimentation we conducted are briefly discussed.